In a crossed static homogeneous electric and magnetic field charged particles describing trochoidal orbits in a plane perpendicular to the direction of the magnetic field are focused, but a beam emitted by a point source in a finite solid angle spreads out indefinitely in the direction parallel to the magnetic field. The essential characteristics of trochoidal orbits can be preserved if the superimposed magnetic and electric fields are two-dimensional and orthogonal, such as derived from a vector potential, say, Ax = A(y, z), Av = Az = O, and a scalar potential Ф(y, z)= const. A(y, z). The focusing properties of such a field combination depend on the distribution of the magnetic field only. Following some general considerations, specific examples of double focusing field distributions are given, and the electron motion in one of them is treated in detail.

Empirical regularities are sought amongst the experimental data relative to the spontaneous fission rates and the nucleon-pair binding energies of the heavy even-even nuclei. Certain regularities are found which have not hitherto been noted; in particular it appears that the isotopic number D is a significant parameter in relation to these various quantities. The speculation is made that possibly the saddle-point deformation of the protons in the nucleus is greater than that of the neutrons. Predictions are made concerning the spontaneous fission rates of the even-even isotopes of thorium.

This paper is concerned with the theory of the probability distribution of the total number of electrons in the avalanche produced by the release of a single electron in the gas of a proportional counter. The disagreement between existing theory and experimental results is discussed and a new theory is proposed, based on the fact that fluctuations in the number of electrons at a given point in the avalanche are accompanied by fluctuations in the average electron energy. This aspect of the problem is incorporated directly into a simple one-dimensional model of the multiplication process, and the resulting distribution function has a mathematical foim in agreement with that observed experimentally.

The fluctuation in the number of electrons predicted by this theory is not constant, but is determined by a parameter which, for large mean values, is essentially the mean fraction of electrons in the avalanche having energies above the ionization energy of the counter gas. Limits on the variation of this parameter are obtained by calculations of the mean values using a particular two-dimensional model, in which electrons are divided into two classes according as they have energies above, or below, the ionization energy. The experimentally observed fluctuation lies within the predicted range and close to the lower limit; it is concluded that there is little scope for improvement in the resolution to be obtained from the conventional type of proportional counter.

Let R be a ring and let ΓR be the Jacobson-Perlis radical. It is shown that the radical of the ring of row-finite matrices over R is the ring of row-finite matrices over ΓR if and only if ΓR is right-vanishing. This is done by extending the results of an earlier paper. One interesting consequence of the theorem is noted.

Nemilov and Pisarevskii (1957) have reported a 390 keV γ-ray in coincidence with neutrons from a Po-Li source, and interpret this as evidence for a level in B10 at 390 keV. A search for confirmatory evidence of this γ–ray has been unsuccessful, and it is estimated that such n–γ coincidences would have been detected in the experiment here described if they involved 8 per cent or more of the neutrons from the Po-Li source.

A study is made of the generalization of the entropic law (xy)(zw)—(xz)(yw) to an identity connecting two operations. It is shown that such an identity is equivalent “in the large” to the condition that the set of endomorphisms with respect to one operation is closed with respect to the other. Furthermore, for such entropic operations, each may be regarded as a generalized endomorphism of the other and various generalizations of elementary properties of endomorphisms are obtained. Examples of entropic pairs of operations are quite common in mathematics and a number of these are discussed. An important aspect of the paper is the matrix notationintroduced to facilitate what would otherwise be extremely cumbersome computations with entropic operations.